# Statistical evaluation of proxies to estimate the erosivity factor of precipitation

### Calculation results and R and PR deviations

The calculation R-the factor of the 28 stations varied between 290.8 MJ mm ha−1 h−1 a−1 and 7153.7 MJ mm ha−1 h−1 a−1 with an average of 2424.1 MJ mm ha−1 h−1 a−1. As shown in Figure 1, all of the proxies examined were found to be surprisingly strongly correlated with the R-factor (r> 0.62, pR -factor in Australia. This finding is at odds with the common notion that regional differences in physiographic parameters preclude a model of the R -factor of application to other fields. The reason could be that the PR the selected models were established in different regions and developed by the Rvalue calculated by the USLE or RUSLE method. The PR value is actually an approximation of the R -calculation of the USLE or RUSLE factor. Therefore, the PR the value is strongly linked to the R value, and the difference between PR and R reflects the correction coefficient, which is the regional difference, and therefore any PR values ​​were strongly correlated with R values.

The linear regression equation of FIG. 1 can be applied to change the gap between the PRand R. Table 3 lists the RMSE , ERM and NSE for the estimate of R directly used values PR and modified PR. The direct estimate showed a high discrepancy, the ERM ranged from 18.3 to 153.0%, the RMSE ranged from 561 to 2279 MJ mm ha−1 h−1 a−1and the NSE ranged from −2.1 to 0.89. Revised calculations, as shown in Table 3, indicated the ERM ranged from 13.1 to 77.4%, the RMSE ranged from 302 to 1482 MJ mm ha−1 h−1 a−1and the NSE ranged from 0.23 to 0.97. The RMSE (p = 0.002) and the ERM(p= 0.01) were significantly reduced, and the NSE (p = 0.009) increased significantly under the correction, which further proved that the error was chain-reduced when the value changed PRhas been used. However, all PR s showed a similar expression, such as the RMSE , MRE,and NSE of PR11, PR12, PR13and PR14which changed slightly after calibration.

### Correlation comparison between R and PRS using Meng’s test

According to the step-by-step testing method devised in this study, the flowchart is shown in Fig. 2, Eq. (9) divided the correlation coefficients by 15 PR the sand R into two classes: L1 (r1, r2, r6, r9), and O1 (r3, r4, r5, rseven, r8, rten, r11, r12, r13, r14, r15), and no significant difference was observed in the L1 to classify (p > 0.06). The correlation coefficients in the O1 class were separated into L2 (r3, r4, r8, rten) and O2 (r5, rseven, r11, r12, r13, r14, r15) classes, and no contrast was detected in the L2 to classify (p> 0.35). The correlation coefficients in the O2 class were divided into L3 (r5, rseven, r12) and O3 (r11, r13, r14, r15) classes, and the coefficients in the L3 class showed statistical equality under the Meng test. The correlation coefficients of PR14 and PR15 were significantly higher than those of r11 and r13, and other statistical tests indicated no significant difference between r11 and r13, r14and r15, respectively. After four rounds (Fig. 2), Meng’s stepwise test procedure ranked the correlation between PRI and R, rIas following:

r_{1 , } = r_{2} = r_{6} = r_{9}

(14)

As expected, the correlation generally became stronger with finer temporal resolution of precipitation data, and daily proxies were more correlated with Rthan monthly ones, which in turn were more strongly correlated with R than annual ones. The ERM , RMSE and NSE (Table 3) showed a very similar rank to the correlation in Eq. (14).

### Recommend PR for rainfall data with different resolutions

Two annual proxies, PR1 and PR2 showed a strong correlation with R(Fig. 1a,b). The t test indicated that the coefficient of PR1 was equal to 1, and the constant term of PR1 was significantly equal to 0. It showed that the Rthe value in Australia can be directly estimated by PR1and Table 3 shows that no striking variation was observed among the ERM, RMSEand NSE when estimating the R revised value PR1. The coefficient and the constant term of PR2 were significantly different from 1 and 0, respectively. She showed that the estimate of the Rvalue in Australia directly by PR2 would result in a significant discrepancy. Based on eq. (14), no statistical difference was observed between r1 and r2. Therefore, we suggest the use of PR1 and revised PR2 as predictors of R -factor when only annual rainfall data are available.

Among the monthly proxies, PR4, PR5and PRseven are based on the modified Fournier index (MFI ), while PR6 is based on the Fournier index (F ) (Table 1). Both indices have been widely applied to estimate the R-factor31,32,33,34. Fox et al.ten and Yue et al. (2014) found that the MFI was more strongly correlated with MAP that the F . Arnoldus35 and Hernando and Romana36 shows a stronger correlation between the MFI and R that between the F and R . The Australian data we used agrees well with the observations above. Although the MFIand F were both strongly correlated with MAPand R (Fig. 3), Meng’s test indicated that the MFI was more correlated with both MAPand R(pMFI-monthly proxies based on, PR4, PR5and PRseveneven showed a stronger correlation with R that PR6 (Eq. (14)), a daily proxy for the R -factor. Therefore, we concluded that the MFI is higher than the F in building a proxy for R-factor. The t-test of the regression equation of PR3PR8 indicated that the coefficients of the equations were significantly different from 1. Table 3 shows that the direct use of the PRs to estimate the R -factor produced large errors. Therefore, when only monthly rainfall data are available, it is best to use the PR5 and PRseven predict the R -factor.

Among the daily proxies, PR14 and PR15 were developed in Australia, and their correlation with Rranked first among 15 proxies. PR11 and PR13two Chinese proxies, ranked second in terms of correlation with R. The reason for the superior performance of PR14 and PR15 may not simply be due to the fact that they were based on Australian data. Capturing periodic variation over the course of a year, the model developed by Yu and Rosewell37 (the original version of PR14 and PR15) performed best among the 8 R-proxies in the northeast of Spainseven.

The t-test of the regression equation of PRtenPR13 indicates that the coefficient and the constant are equal to 1 (p> 0.06) and 0 (p> 0.33), respectively. Table 3 shows that no notable improvement was observed in the difference between Rvalue when estimated by the revised equations. Therefore, PRtenPR13 can be used to estimate the Rassess. The coefficient and the constant term of PR9 were significantly different from 1 and 0, respectively. The revised version PR9 was recommended when estimating the R-factor in Australia. The coefficient of PR14 was significantly equal to 1 (p= 0.87) and the constant was significantly greater than 0 (p= 0.02), indicating that the equation for PR14 in Fig. 1k probably leads to a large deviation in the estimate Rvalues ​​for dry areas. The constant term of PR15 was equal to 0 (p= 0.45) and the coefficient was significantly greater than 1 (p= 0.00), indicating that PR15 underestimated the Rassess. PR15 was established using the RUSLE model, which may have led to this result. Nearing et al. (2017) concluded that RUSLE underestimated 14%, compared to RUSLE2 and USLE, in estimating the R-factor. This result coincided with the coefficient of 0.19, which is shown in Fig. 1l. We suggest using the revised version PR14 and PR15 as substitutes for R-factor when daily rainfall data is available. However, with ERMs of 26.1% and 23.6%, respectively (Fig. 1i,j), the accuracy remains satisfactory using PR11 and PR13 as a substitute for R-factor.